Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1995 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1995 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,21)(13,25)(16,27)(18,29)(20,30)(23,26)(24,28)(31,33)(32,35)(34,36);(1,2)(3,4)(5,7,10,18)(6,11,14,26)(8,16,20,32)(9,23,25,17)(12,27,28,35)(13,29,22,15)(19,31,30,36)(21,34,24,33);(2,4)(8,20)(11,26)(12,28)(13,22)(16,32)(21,24)(31,36)(33,34);(5,8,10,20)(6,12,14,28)(7,16,18,32)(9,24,25,21)(11,27,26,35)(13,30,22,19)(15,31,29,36)(17,34,23,33)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72306393, #72306405, #72306414, … |
| maximal normal subgroup of | #72306393, #72306405, #72306414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407036);